Difference between revisions of "1991 AHSME Problems/Problem 19"

(Solution 4 (Really simple similar triangles))
(Solution 4 (Really simple similar triangles))
 
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We have <math>\angle ABC = \arcsin\left(\frac{3}{5}\right)</math> and <math>\angle DBA=\arcsin\left(\frac{12}{13}\right).</math> Now we are trying to find <math>\sin(\angle DBE)=\sin\left(180^{\circ}-\angle DBC=\sin(180^{\circ}-\arcsin\left(\frac{3}{5}\right)\right)-\arcsin\left(\frac{12}{13}\right)=\sin(\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{12}{13}\right)).</math> Now we use the <math>\sin</math> angle sum identity, which states <math>\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).</math> Using this identity yields <math>\sin\left(\arcsin\left(\frac{3}{5}\right)\right)\cos\left(\arcsin\left(\frac{12}{13}\right)\right)+\cos\left(\arcsin\left(\frac{3}{5}\right)\right)+\sin\left(\arcsin\left(\frac{12}{13}\right)\right).</math>
 
We have <math>\angle ABC = \arcsin\left(\frac{3}{5}\right)</math> and <math>\angle DBA=\arcsin\left(\frac{12}{13}\right).</math> Now we are trying to find <math>\sin(\angle DBE)=\sin\left(180^{\circ}-\angle DBC=\sin(180^{\circ}-\arcsin\left(\frac{3}{5}\right)\right)-\arcsin\left(\frac{12}{13}\right)=\sin(\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{12}{13}\right)).</math> Now we use the <math>\sin</math> angle sum identity, which states <math>\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).</math> Using this identity yields <math>\sin\left(\arcsin\left(\frac{3}{5}\right)\right)\cos\left(\arcsin\left(\frac{12}{13}\right)\right)+\cos\left(\arcsin\left(\frac{3}{5}\right)\right)+\sin\left(\arcsin\left(\frac{12}{13}\right)\right).</math>
 
== Solution 4 (Really simple similar triangles)==
 
By Pythagoras on right triangle <math>ABD</math>, <math>BD = 13</math>. Draw the line through <math>D</math> parallel to <math>CE</math>. Let this line intersect <math>AC</math> at <math>F</math>.
 
[asy]
 
unitsize(0.3 cm);
 
 
pair A, B, C, D, E, F;
 
 
A = (0,3);
 
B = (4,0);
 
C = (0,0);
 
D = scale(12/5)*rotate(90)*(B - A) + A;
 
E = (D + reflect(B,C)*(D))/2;
 
F = extension(A,C,D, D + C - E);
 
 
draw(A--C--E--D);
 
draw(A--B--D--cycle);
 
draw(A--F--D);
 
 
label("<math>A</math>", A, W);
 
label("<math>B</math>", B, S);
 
label("<math>C</math>", C, SW);
 
label("<math>D</math>", D, NE);
 
label("<math>E</math>", E, SE);
 
label("<math>F</math>", F, NW);
 
[/asy]
 
Since <math>\angle DAB = 90^\circ</math>, <math>\angle FAD + \angle CAB = 90^\circ</math>. Hence, right triangles <math>ABC</math> and <math>DAF</math> are similar. Then
 
<cmath>\frac{AF}{4} = \frac{12}{5},</cmath>
 
so <math>AF = 48/5</math>.
 
 
Since quadrilateral <math>CEDF</math> is a rectangle, <math>DE = CF = AC + AF = 3 + 48/5 = 63/5</math>. Therefore,
 
<cmath>\frac{DE}{DB} = \frac{63/5}{13} = \frac{63}{65}.</cmath>
 
Then <math>m = 63</math>, <math>n = 65</math>, and <math>m + n = 63 + 65 = \boxed{128}</math>. The answer is (B).
 
 
~ math31415926535
 
  
 
== See also ==
 
== See also ==

Latest revision as of 21:41, 23 November 2020

Problem

[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]

Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If \[\frac{DE}{DB}=\frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers, then $m+n=$

$\text{(A) } 25\quad \text{(B) } 128\quad \text{(C) } 153\quad \text{(D) } 243\quad \text{(E) } 256$

Solution 1

Solution by e_power_pi_times_i


Let $F$ be the point such that $DF$ and $CF$ are parallel to $CE$ and $DE$, respectively, and let $DE = x$ and $BE^2 = 169-x^2$. Then, $[FDEC] = x(4+\sqrt{169-x^2}) = [ABC] + [BED] + [ABD] + [AFD] = 6 + \dfrac{x\sqrt{169-x^2}}{2} + 30 + \dfrac{(x-3)(4+\sqrt{169-x^2})}{2}$. So, $4x+x\sqrt{169-x^2} = 60 + x\sqrt{169-x^2} - 3\sqrt{169-x^2}$. Simplifying $3\sqrt{169-x^2} = 60 - 4x$, and $1521 - 9x^2 = 16x^2 - 480x + 3600$. Therefore $25x^2 - 480x + 2079 = 0$, and $x = \dfrac{48\pm15}{5}$. Checking, $x = \dfrac{63}{5}$ is the answer, so $\dfrac{DE}{DB} = \dfrac{\dfrac{63}{5}}{13} = \dfrac{63}{65}$. The answer is $\boxed{\textbf{(B) } 128}$.

Solution 2

Solution by Arjun Vikram

Extend lines $AD$ and $CE$ to meet at a new point $F$. Now, we see that $FAC\sim FDE \sim ACB$. Using this relationship, we can see that $AF=\frac{15}4$, (so $FD=\frac{63}4$), and the ratio of similarity between $FDE$ and $FAC$ is $\frac{63}{15}$. This ratio gives us that $\frac{63}5$. By the Pythagorean Theorem, $DB=13$. Thus, $\frac{DE}{DB}=\frac{63}{65}$, and the answer is $63+65=\boxed{\textbf{(B) } 128}$.


Solution 3 (Trig)

We have $\angle ABC = \arcsin\left(\frac{3}{5}\right)$ and $\angle DBA=\arcsin\left(\frac{12}{13}\right).$ Now we are trying to find $\sin(\angle DBE)=\sin\left(180^{\circ}-\angle DBC=\sin(180^{\circ}-\arcsin\left(\frac{3}{5}\right)\right)-\arcsin\left(\frac{12}{13}\right)=\sin(\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{12}{13}\right)).$ Now we use the $\sin$ angle sum identity, which states $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b).$ Using this identity yields $\sin\left(\arcsin\left(\frac{3}{5}\right)\right)\cos\left(\arcsin\left(\frac{12}{13}\right)\right)+\cos\left(\arcsin\left(\frac{3}{5}\right)\right)+\sin\left(\arcsin\left(\frac{12}{13}\right)\right).$

See also

1991 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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